Theorem 3.20 in Section 3.4 determines the conditions on the parameter set of the bivariate
powered exponential model which guarantee the positive definiteness in Rn, n ∈ {1, 3}. These
conditions set the boundaries for ρLC depending on given α11, α22, α12, s11, s22, s12. Moreover,
parts (i)-(iv) of Theorem 3.20 state that the infimum in (3.22) can be positive only if α12 ≥
max{α11, α22}. These restrictions lead to the correlated maximum likelihood estimates. In
order to reduce the correlation of estimates and increase the speed of the parameters search, we reparametrise the model inside the RandomFields package, following the approach for the
bivariate Mat´ern model. We introduce auxiliary parameters βred, ρredand ρmax such that
α12= max{α11, α22} + βred(2 − max{α11, α22}),
where βred∈ [0, 1] and
ρ = ρredρmax,
where |ρred| ∈ [0, 1] and ρmax = α11α22sα1111s
α22 22 /(α212s 2α12 12 ) infr>0g(r) with g(r) = " rα11+α22−2α12e2(s12r)α12−(s11r)α11−(s22r)α22q (n) α11,s11(r)q (n) α22,s22(r) (q(n)α12,s12(r))2 # .
When performing loglikelihood optimization we vary the parameters α11, α22, βred, ρred, s11,
s22, s12and based on their values compute α12and ρ. The parameters βredand ρredare chosen
independently of α11, α22, α12, s11, s22, s12. With this reparametrization any combination of
values of α11∈ (0, 1], α22∈ (0, 1], βred∈ [0, 1], s11, s22, s12> 0, ρred∈ [−1, 1] leads to the valid
bivariate powered exponential covariance model.
The calculation of ρmax requires finding an infimum of the function g(r). We are interested
in positive values of ρmax or, in other words, in cases (i)-(iv) of Theorem 3.20. Consider the
behaviour of the function g(r) at zero and at infinity provided that the conditions (i)-(iv) in Theorem 3.20 hold true. Then we have
lim
5.2. Implementation details 65
Under condition (i) of Theorem 3.20 we have lim
r→0g(r) = 1.
Conditions (ii) - (iv) of Theorem 3.20 yield lim
r→0g(r) = ∞.
Thus, if the one of the conditions (ii) - (iv) of Theorem 3.20 holds, the function g attains its minimum in (0, ∞) and if the conditions (i) of Theorem 3.20 holds, g attains its minimum in [0, ∞). In general g is not unimodal on [0, ∞), therefore we use the following heuristic algorithm to find the minimum. First we locate an interval which is likely to contain a global minimum.
To do so we choose the starting points rm = 10k, k ∈ {−10, −9, . . . , 10}, and repeat Algorithm
1 for each k or until gmin = 0 for some k. If for some k ∈ {−10, −9, . . . , 10} Algorithm 1
returned gmin = 0, we stop and set ρmax = 0. If gmin = −1, we located an interval with a local
maximum. We run again Algorithm 1 with a new starting point rm/2. If gmin > 0 for several
k ∈ {−10, −9, . . . , 10}, we apply the golden-section search algorithm (Press et al., 1982) to the corresponding intervals in order to find the minimum. Then the smallest minimum is the value
of ρmax. We noticed that in the intervals, where g takes values less than 0.05, the algorithm
often stops before finding a minimum. This happens when the function does not decrease fast enough in the neighborhood of the true minimum and the golden-section search algorithm fails to locate it, since its precision is limited, see Press et al. (1982) for more details. To avoid these
situations, we set ρmax = 0 if we came across a point r which guarantees that ρmax ≤ 0.05 while
running Algorithm 1 or the golden-section search algorithm. From a practical perspective this is not a strict restriction, since the use of the bivariate model with such a low cross-correlation parameter is superfluous. For the sake of consistency, we choose ε = 0.05 in Algorithm 1.
66 Chapter 5. Data analysis with bivariate covariance models
Compute rl= rm/2, rr= 2rm. gmin= 0.
while g(rm) ≥ min{g(rr), g(rl)} and min{g(rm), g(rl), g(rr)} > ε do
if g(rm) ≥ max{g(rr), g(rl)} then rm= rl ; gmin = −1; break; end if g(rl) ≤ g(rm) then rm= rl; gmin = g(rm) = g(rl); rl= rl/2; end if g(rr) ≤ g(rm) then rm= rr; gmin = g(rm) = g(rr); rr= 2rr; end if min{g(rm), g(rl), g(rr)} ≤ ε then rr= rl= 0; gmin = 0; end end return gmin, rl, rr;
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Hiermit erkl¨are ich, dass ich die vorliegende Arbeit mit dem Titel “Bivariate Gaussian ran-
dom fields: models, simulation, and inference“ selbstst¨andig angefertigt und keine anderen als
die angegebenen Hilfsmittel verwendet habe.
Mannheim, den 28.05.2018 . . . .